Projection Definition and 434 Threads

This is not really a homework question per se but I wasn't sure where else to put it: In a script I'm reading the following set is defined: P(n)_k := \{p \in S(n) | p^2 = p, \text{trace } p = k\} (i.e. the set of all real orthogonal projection matrices with trace k). Now the following...

Homework Statement The following are all vectors: A = <2, 1, 1> B = <1, -2, 2> C = <3, -4, 2> Find the projection of (A + C) in the direction of B Homework Equations Dot product? The Attempt at a Solution I was not sure what the meant in this question. I added A and C...

Homework Statement Given a plane \Pi with normal n=i-2j+k and a vector v=3i+4j-2k calculate the projection of v onto \Pi and the reflection of v with respect to \Pi. The Attempt at a Solution I need to check that I'm doing this is right. I think I need v - (v \cdot n)n =...

I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates...

If V is a vectorspace and v is a vector in V. will the projection of v onto V be v?

I'm doing practice problems for my exam, but I don't really know how to get this one. I'd like to just be able to understand it before my test if anyone can help explain it! Prove from the definition of projection (given below) that if projv=u(sub A) then A=A^2 and A=A^T. (Hint: for the...

Given sets X1,...,Xn the projection from the product X1 * ... *Xn is the function pri = prxi : X1 * ... *Xn → Xi , pri(t1, ..., ti, ...,tn) := ti Im having a hard time figuring out what exactly is going on here. Some coordinate from all sets X1 to Xn are getting multiplied together? The...

Homework Statement Think of the following matrix A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) as a transformatiom of \mathbb{R}^3 onto itself. Describe A as a projection onto a plane followed by a shearing motion of the plane. 2...

Hi PF members, where I can find application notes for making a small Holographic Projector? I am looking for a schematic diagram of a small, compact and reliable Holographic Projector, based on LED or Laser emitter. I am completely new on this application, so let be as more exhaustive as...

I found a final answer online, but my vector is slightly different. I haven't been able to catch my mistake. I'm supposed to find the orthogonal projection of the given vector on the given subspace W of the the inner product space V. P1 has dimension 2 and basis = {1,x}...

Homework Statement Let (X,||\cdot||) be a normed vector space and suppose that Y is a closed vector subspace of X. Show that the map ||x||_1=\inf_{y \in Y}||x-y|| defines a pseudonorm on X. Let (X/Y,||\cdot||_1) denote the normed vector space induced by ||\cdot||_1 and prove that the...

Hey guys, I've often seen in the definition of a Fiber bundle a projection map \pi: E\rightarrow B where E is the fiber bundle and B is the base manifold. This projection is used to project each individual fiber to its base point on the base manifold. I then see a lot of references to...

Ket Notation -- Effects of the Projection Operator Homework Statement From Sakurai's Modern Quantum Mechanics (Revised Edition), it is just deriving equation 1.3.12. Homework Equations \begin{eqnarray*}\langle \alpha |\cdot (\sum_{a'}^N |a'\rangle \langle a'|) \cdot|\alpha \rangle...

Homework Statement A projectile is launched with a speed v at an angle theta above the horizontal. Ignore air resistance. Derive an expression for the angle theta in terms of the parameters of the problem such that the horizontal distance from the launch of the object is N times greater than...

Homework Statement A particle is projected from a point P on an inclined plane, up the line of greatest slope through P, with initial speed V. The angle of the plane to the horizontal is θ. (i) If the plane is smooth, and the particle travels for a time \frac{2Vcosθ}{g} before coming...

Using a scalar projections how do you show that the distance from a point P(x1,y1) to line ax + by + c = 0 is \frac{|ax1 +b y1 + c|}{\sqrt{a^2 +b^2}} I do not know how to approach this, please provide some guidance.

I am working on an implementation of the Gilbert–Johnson–Keerthi distance algorithm and am having difficulty with some of the more general math involved. I am able to find the projection of a point onto a plane because I'm given at least three points on the plane and the point that is to be...

Dear all, I have a question concerning Depth of Field –*I'm trying to find a depth of field calculation method that applies for video projection. Background info is that I often have projection on non-planar surfaces and like to find a method that allows to calculate (without trying out) if...

I just want to make sure my thinking is correct with a problem I'm working on. I'm trying to write a function that will take a point on a plane above a sphere, and then project it onto that sphere. From there project the point onto the x,y plane by following the normal vector of the sphere I...

Hello everybody, I am working on a project for merging the images of multiple meteorology radars covering a large area. I would like to project each of the images onto a spherical surface and merge them. Do you know if there is a piece of software that can help me with that? I can develop in...

So my book says Lets suppose, We have two vector v and u w=projection of u ev= unit vector θ=angle between the two w=(u.ev)ev or w=( (u.v)/(v.v) )v Now, the second equation is fairly easy to understand if we understand the first one because ev= v / |v| What is...

Homework Statement a.) A hiker is climbing a mountain whose height is z = 1000 - 2x**2 - 3y**2. When he is at the point (1,1,995) in what direction should he move in order to ascent as rapidly as possible? b.) If he continues along a path of steepest ascent, obtain the equation of the curve...

Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...

Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...

How do I find a vector which is the projection of another vector onto a plane? By projection, I mean perpendicular projection onto this plane. I know that this vector must lie in the plane and have a minimum angle with the original vector, but it seems like setting up the problem in this...

8) U=\{x=(x_{1},x_{2},x_{3},x_{4})\in R^{4}|x_{1}+x_{2}+x_{4}=0\} is a subspace of R^{4} v=(2,0,0,1)\in R^{4} find u_{0}\in U so ||u_{0}-v||<||u-v|| how i tried: U=sp\{(-1,1,0,0),(-1,0,0,1),(0,0,1,0)\} i know that the only u_{0} for which this innequality will work is if it will be the...

7) T:R^{2}->R^{2} projection transformation on X-axes parallel to the line y=-\sqrt{3}x find the representative matrices of T{*} by B=\{(1,0),(0,1)\} basis how i tried: i understood that the x axes stayed the same but the y axes turned into y=-\sqrt{3}x our T takes some vector and...

Homework Statement Solve du/dt=Pu when P is a projection. [1/2 1/2 = du/dt with 1/2 1/2] [5 = u(0). 3] Part of u(0) increases exponentially while the nullspace part stays fixed. Homework Equations du/dt = Au with u=u(0) at t=0 The Attempt at a...

Homework Statement Let 2 be the plane containing the line, l:(x,y,z)= t(6,4,2)+ (3,-4,2) and the point Q(5, -7, 7). Let P be the point (-6, -12,5) a) Find the projection of QP onto 2. Homework Equations I know the projection eqn would be (( plane 2 dot QP)/ magnitude on QP) * components...

Homework Statement Let u=(-1,-2,-2,2) and v=(-1,-2,-2,-1) and let V=span{u,v}. (Just to be clear, u and v are column vectors) Find the standard matrix that projects points (orthogonally) onto V.Homework Equations The Attempt at a Solution I started by making a matrix A=[u,v], which...

If we have a matrix M with a kernel, in many cases there exists a projection operator P onto the kernel of M satisfying [P,M]=0. It seems to me that this projector does not in general need to be an orthogonal projector, but it is probably unique if it exists. My question: is there a standard...

Homework Statement If A and B are sets, prove that a subset \Gamma\subset A X B is the graph of some function from A to B if and only if the first projection \rho: \Gamma\rightarrow A is a bijection. Homework Equations The Attempt at a Solution I first thought that i should...

Homework Statement Parallelograms 5cm side is located on a plain alpha and its 3cm side forms an angle with the plain equal to 30 degrees. Find the area for the parallelograms projection on the plain alpha, if an angle in the original parallelogram is 60 degrees. Homework Equations all...

I'm a little confused about the relaionship between susy transformations involving Q generators & gso projection in superstring theory. If gso projection is used in RNS sectors, does that only eliminate certain states, such as tachyon, then susy must still be applied? Or does gso projection...

ok i have a problem to work on in my new course, and i was wondering what i need to do to tackle it. the question is as follows: An electron is projected with an initial velocity Vo=10^7m/s into the uniform field between the parallel plates "E". the direction of the field is vertically...

Homework Statement Consider the two vectors A=ai and B=3i+4j. What must be the value of a if the component of A along B is 6? Homework Equations The Attempt at a Solution I've arrived at the correct answer by finding the angle between the x component of B (3) and B itself which...

I have vector [ tex ] v [ /tex ] produced by an orthogonal projection of vector [ tex ] w [ /tex ] over plane spanned by vectors [ tex ] u_1 [ /tex ] and [ tex ] u_2 [ /tex ], in a three dimensional space. If I know [ tex ] v [ /tex ], [ tex ] u_1 [ /tex ] and [ tex ] u_2 [ /tex ], how could I...

I imagine differential equations and statistics are used. What type of math is used to predict the path of hurricanes?

EDIT: I am looking for a correction in my understanding of relativity. A model of the universe as it appears to a photon will be presented in a LOGICALLY RIGOROUS MANNER based on my understanding of physics by examining the limiting effect as one approaches the speed of light. Since my...

When it comes to graphics (what I have learned) is that what you see on the screen is data placed in a memory (video buffer) and then the data or pixel is then plotted onto the screen. So when you manipulate the data in the video buffer you will also manipulate the pixels on the screen. Then...

Homework Statement the given vector v and subspace W. (a) Let W be the subspace with basis {(1 1 0 1)T, (0 1 1 0)T, (-1 0 0 1)T} and v = (2 1 4 0)T. Homework Equations ProjWv = (<W, v> / <W, W>) * W The Attempt at a Solution So I'm trying to wrap my head around this...

Hello, Is it possible to project a hypersphere (a 3-sphere) onto a plane? is this possible using stereographic projection? Please, if this is possible I would appreciate a nice explain me about how to do it. Thank you! Carol

Given x=(2,-5,4)^T and y=(1,2,-1)^T

Why is the Trace of a projection is its Rank. Thank you

Hello, I was wondering if anyboday can clarify this for me. I am trying to project a sphere into a plane, I am using the stereogriphic projection which I believe in cartesian coordinates is: x'=x/(R^2-z) y'=y/(R^2-z) where x' and y' are the coordinates in the plane, (x,y,z) the...

Homework Statement Given \pi: \begin{cases}l_1: \frac{x-2}{4}\ = \frac{y-1}{2}\ = \frac{z+5}{-4} &\\l_2: \frac{x+4}{-2}\ = \frac{y+1}{0}\ = \frac{z}{1} & \end{cases} and the point M=(1,2,3) outside the plane. Find the projection of the line M_1P on plane \pi where P is the intersection point...

can someone explain physically why escape velocity is independent of angle of projection.

Homework Statement Minimize ||cos(2x) - f(x)|| where f(x) is a a function in the span of {(1,sin(x),cos(x)} Where the inner produect is defined (1/pi)(integral from -pi to pi of f(x)g(x) dx) Homework Equations I found f(x) to be zero. Is this correct I am uneasy about this...

I am trying to project a voxelized cube. I have created a cube of size 255x255x255 with intensity values one. I have derived the perspective transformation matrix and constructed the projection-matrix. I have implemented the projection of a cube using voxel-driven projection algorithm in...

Homework Statement Let {M_i} be an orthogonal sequence of complete subspaces of a pre-Hilbert space V, and let P_i be the orthogonal projection on M_i. Prove that {P_i(e)} is Cauchy for any e in V 2. The attempt at a solution I'm trying to prove as n and m goes infinity...